The tangent line to a plane curve at a given point is the [[Linear Function|straight line]] that just touches the curve at that point. - The slope of a tangent line to a curve at a point is the slope of the curve at the point. ### Formal Definition The tangent line to the curve $y = f(x)$ at the point $P(a, f(a))$ is the line through a pair of [[Limit|infinitely close]] points, one at $P$ and one near $P$, with the slope $ m = \lim_{x \to a}{\frac{f(x) - f(a)}{x - a}} = \lim_{h \to 0}{\frac{f(x + h) - f(x)}{h}} $ provided that this limit exists.